Controllability of nonlinear PDE’s: Agrachev–Sarychev approach

Armen Shirikyan[1]

  • [1] CNRS (UMR 8088), Département de Mathématiques Université de Cergy–Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin 95302 Cergy–Pontoise Cedex, France

Journées Équations aux dérivées partielles (2007)

  • page 1-11
  • ISSN: 0752-0360

Abstract

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This short note is devoted to a discussion of a general approach to controllability of PDE’s introduced by Agrachev and Sarychev in 2005. We use the example of a 1D Burgers equation to illustrate the main ideas. It is proved that the problem in question is controllable in an appropriate sense by a two-dimensional external force. This result is not new and was proved earlier in the papers [AS05, AS07] in a more complicated situation of 2D Navier–Stokes equations.

How to cite

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Shirikyan, Armen. "Controllability of nonlinear PDE’s: Agrachev–Sarychev approach." Journées Équations aux dérivées partielles (2007): 1-11. <http://eudml.org/doc/10631>.

@article{Shirikyan2007,
abstract = {This short note is devoted to a discussion of a general approach to controllability of PDE’s introduced by Agrachev and Sarychev in 2005. We use the example of a 1D Burgers equation to illustrate the main ideas. It is proved that the problem in question is controllable in an appropriate sense by a two-dimensional external force. This result is not new and was proved earlier in the papers [AS05, AS07] in a more complicated situation of 2D Navier–Stokes equations.},
affiliation = {CNRS (UMR 8088), Département de Mathématiques Université de Cergy–Pontoise, Site de Saint-Martin, 2 avenue Adolphe Chauvin 95302 Cergy–Pontoise Cedex, France},
author = {Shirikyan, Armen},
journal = {Journées Équations aux dérivées partielles},
keywords = {Burgers equation; approximate controllability; exact controllability in projection; Agrachev–Sarychev method},
language = {eng},
month = {6},
pages = {1-11},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Controllability of nonlinear PDE’s: Agrachev–Sarychev approach},
url = {http://eudml.org/doc/10631},
year = {2007},
}

TY - JOUR
AU - Shirikyan, Armen
TI - Controllability of nonlinear PDE’s: Agrachev–Sarychev approach
JO - Journées Équations aux dérivées partielles
DA - 2007/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 11
AB - This short note is devoted to a discussion of a general approach to controllability of PDE’s introduced by Agrachev and Sarychev in 2005. We use the example of a 1D Burgers equation to illustrate the main ideas. It is proved that the problem in question is controllable in an appropriate sense by a two-dimensional external force. This result is not new and was proved earlier in the papers [AS05, AS07] in a more complicated situation of 2D Navier–Stokes equations.
LA - eng
KW - Burgers equation; approximate controllability; exact controllability in projection; Agrachev–Sarychev method
UR - http://eudml.org/doc/10631
ER -

References

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  1. A. A. Agrachev and Yu. L. Sachkov, Control Theory from Geometric Viewpoint, Springer-Verlag, Berlin, 2004. Zbl1062.93001MR2062547
  2. A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108–152. Zbl1075.93014MR2127744
  3. —, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. 265 (2006), no. 3, 673–697. Zbl1105.93008MR2231685
  4. —, Solid controllability in fluid dynamics, Instabilities in Models Connected with Fluid Flow. I (C. Bardos and A. Fursikov, eds.), Springer, 2007, pp. 1–35. MR2459254
  5. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. Zbl0189.40603
  6. S. S. Rodrigues, Navier-Stokes equation on the rectangle: controllability by means of low mode forcing, J. Dyn. Control Syst. 12 (2006), no. 4, 517–562. Zbl1105.35085MR2253360
  7. —, Controllability of nonlinear PDE’s on compact Riemannian manifolds, Workshop on Mathematical Control Theory and Finance, vol. Lisbon, 10–14 April, 2007, pp. 462–493. 
  8. A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. 266 (2006), no. 1, 123–151. Zbl1105.93016MR2231968
  9. —, Exact controllability in projections for three-dimensional Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 4, 521–537. Zbl1119.93021MR2334990
  10. M. E. Taylor, Partial Differential Equations. I–III, Springer-Verlag, New York, 1996-1997. Zbl0869.35003MR1395148

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